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Theorem 19.45 2226
Description: Theorem 19.45 of [Margaris] p. 90. See 19.45v 1989 for a version requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
Hypothesis
Ref Expression
19.45.1 𝑥𝜑
Assertion
Ref Expression
19.45 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Proof of Theorem 19.45
StepHypRef Expression
1 19.43 1877 . 2 (∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
2 19.45.1 . . . 4 𝑥𝜑
3219.9 2193 . . 3 (∃𝑥𝜑𝜑)
43orbi1i 911 . 2 ((∃𝑥𝜑 ∨ ∃𝑥𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
51, 4bitri 274 1 (∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wo 845  wex 1773  wnf 1777
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-12 2166
This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1774  df-nf 1778
This theorem is referenced by:  eeorOLD  2324
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